A new lower bound for the chromatic number of general Kneser hypergraphs

Sani, Roya Abyazi; Alishahi, Meysam

doi:10.48550/arxiv.1704.07052

Cited by 1 publication

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“…Tucker's combinatorial lemma [32] and Fan's combinatorial lemma [15] are two powerful tools in combinatorial topology. The problem of finding a lower bound for the chromatic number of general Kneser hypergraphs via Tucker's lemma [32] and Fan's lemma [15] has been extensively studied in the literature, see [1,2,3,6,7,8,14,21,23,24,25,27,29,30,31,33,34]. The following lemma was proposed by Meunier [24].…”

Section: Topological Lower Bound Of χ Kg P (N K) S-stab For Prime Pmentioning

confidence: 99%

On the chromatic number of almost s-stable Kneser graphs

Chen¹

2017

Preprint

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In 2011, Meunier conjectured that for positive integers n, k, r, s with k ≥ 2, r ≥ 2, and n ≥ max({r, s})k, the chromatic number of sstable r-uniform Kneser hypergraphs is equal to n−max({r,s})(k−1) r−1 . It is a strengthened version of the conjecture proposed by Ziegler (2002), andAlon, Drewnowski andLuczak (2009). The problem about the chromatic number of almost s-stable r-uniform Kneser hypergraphs has also been introduced by Meunier (2011).For the r = 2 case of the Meunier conjecture, Jonsson (2012) provided a purely combinatorial proof to confirm the conjecture for s ≥ 4 and n sufficiently large, and by Chen (2015) for even s and any n. The case s = 3 is completely open, even the chromatic number of the usual almost s-stable Kneser graphs.In this paper, we obtain a topological lower bound for the chromatic number of almost s-stable r-uniform Kneser hypergraphs via a different approach. For the case r = 2, we conclude that the chromatic number of almost s-stable Kneser graphs is equal to n − s(k − 1) for all s ≥ 2. Set t = n − s(k − 1). We show that any proper coloring of an almost s-stable Kneser graph must contain a completely multicolored complete bipartite subgraph K ⌈ t 2 ⌉⌊ t 2 ⌋ . It follows that the local chromatic number of almost s-stable Kneser graphs is at least t 2 + 1. It is a strengthened result of Simonyi and Tardos (2007), and Meunier's (2014) lower bound for almost s-stable Kneser graphs.

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Section: Topological Lower Bound Of χ Kg P (N K) S-stab For Prime Pmentioning

confidence: 99%

On the chromatic number of almost s-stable Kneser graphs

Chen¹

2017

Preprint

View full text Add to dashboard Cite

show abstract

A new lower bound for the chromatic number of general Kneser hypergraphs

Cited by 1 publication

References 20 publications

On the chromatic number of almost s-stable Kneser graphs

On the chromatic number of almost s-stable Kneser graphs

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